A subroutine modeling a geothermal heat pump unit is executed within each time step of the ground loop simulation. The performance data is supplied with COP’s for different fluid volume flows and entering water temperatures. A crude model could be developed using this data although this would neglect the indoor temperature, air flow, and antifreeze concentration factors. The most accurate model possible must include these factors and that is why the COP will be dissected into its constituents for a complete
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correlation study. That is to say, equations for the capacity and power are individually studied. The hourly Energy Efficiency Ratio (EER) value can be calculated and converted to a COP value for rated conditions using the following equations for cooling and heating respectfully,
(31)
and (32)
where
is the Gross Cooling Capacity (Mbtuh) of unit number u is the compressor power (kW) of unit number u
is the Gross Heating Capacity (Mbtuh) of unit number u is the compressor power (kW) of unit number u
The multiplying constant is a unit conversion from EER to COP.
To develop an equation for cooling capacity, the data is plotted versus the fluid volume flow in for all eight entering water temperatures (EWT) provided in the performance data. A plot of each curve for a 3 ton unit can be seen in Figure 8. It is important to note that the capacity data is in English units while all other data is in metric units. This was done to easily check the gross capacity and compressor power calculations to the performance data, while also being necessary to calculate the EER properly.
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Figure 8: Cooling capacity vs. fluid flow for different entering water temperatures (Trane November 2010).
Each of the curves can now be described as a second order quadratic equation taking the form
. (33)
It is recognizable that the curve and slope of each of the different sets of data appears to be somewhat constant. The coefficient from equation (33)is then plotted versus the entering water temperature for every heat pump unit size. A second order polynomial is then fit to the data and the curve describing the 3 ton unit number 8 can be seen in Figure 9.
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Figure 9: Coefficient A vs. entering water temperature for 3 ton unit #8.
The coefficient can now be written as
(34)
where EWT is the entering water temperature in Celsius. The coefficients and
for all eighteen heat pump sizes can be found in the appendix. This coefficient describes how much the data curves in Figure 8, as the volume flow changes. At lower EWT’s, the coefficient has larger magnitudes suggesting that the capacity is changing more with volume flow. The behavior of coefficient at higher EWT’s suggests that the cooling capacity is dominated more by the water temperature than the volume flow.
The next coefficient to describe the cooling capacity in equation (31), is the linear term . Plotting each of the coefficients versus the respective entering water
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Figure 10: Coefficient B vs. entering water temperature for 3 ton unit #8.
The coefficient can now be written as
(35)
where again, the coefficients , and for all eighteen heat pump sizes can be found in the appendix. The behavior of coefficients and appear to be mirror images of each other and somewhat sporadic. The behavior of coefficient is describing the slope of the curve from Figure 8. The slope at higher temperatures has decreased, suggesting that the cooling capacity becomes more dependent of the EWT than the volume flow at higher temperatures.
The final coefficient describing the cooling capacity is the constant term, . This term is what truly dominates the equation and after performing a second order regression, Figure 11 shows the correlation between coefficient and EWT.
0 10 20 30 40 50
2000 2500 3000 3500 4000 4500 5000 5500
Coefficient B Mbtuh/(m3 /sec)
Entering water temperature (°C) Coefficient B
2nd order fit
31
Figure 11: Coefficient C vs. entering water temperature for 3 ton unit #8.
The coefficient can now be written as
(36)
and plugging into the original polynomial produces one equation for the cooling capacity
(37)
. When plugging in the values for the coefficients
and for unit #8, and using the rated volume flow of 0.0005299
, and an entering water temperature of 25 °C, the cooling capacity is calculated to be
0 10 20 30 40 50
28 30 32 34 36 38 40
Coefficient C (Mbtuh)
Entering water temperature (°C)
Coefficient C 2nd order fit
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35.66 Mbtuh. The supplied performance data shows the cooling capacity of unit number 8 at 25°C (77°F) to be 35.7 Mbtuh at the rated volume flow. Acceptable volume flows for use with these curves are available in the appendix.
Like the cooling capacity first described in equation (33), the heating capacity, cooling compressor power, and heating compressor power are described as follows for all 18 units studied,
, (38)
, (39)
and
. (40)
After trend studies of the coefficients were completed in the same manner as the cooling capacity trend studies above, the correlated equations for , and are developed as follows for any unit 1 through 18
(41)
(42)
(43)
.
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Using the rated value for the volume flow, and coefficients for unit 8, is calculated to be 2.579 kW. Using the values calculated for and and plugging into equation (31) gives,
(44)
The COP for unit 8 at the rated volume flow and an entering fluid temperature of 25 °C published in the performance data is 4.053. The is calculated in the same way using equation (32) with equation (42) and equation (43). The COP for heating and cooling are plotted in Figure 12 using entering water temperatures from the performance data with the unit rated volume flow. The performance data used was not extrapolated past the published EWT's.
Figure 12: COP for 3 ton unit number 8 at rated volume flow (Trane November 2010).
Using second order polynomials for each coefficient describes the capacities and compressor power well enough to avoid error propagation through to the COP calculation. Evaluating all 18 units in the same manner revealed the largest error to be no
0 10 20 30 40
34 temperature (EAT) over the heat exchanger, and the percent concentration of antifreeze in the working fluid. The previous calculations were all performed at the manufacturers rated air volume flow, air temperatures, and using water as the working fluid. The correction factors for capacities and compressor power as a function of the EAT are plotted in Figure 13. The rated EAT can be seen where the correction factor is equal to one. It is important to note that the EAT for cooling is the wet bulb temperature while for heating it is the dry bulb temperature. Calculation of the wet bulb temperature is discussed in section 4.4.
Figure 13: Correction factors for entering air temperature (Trane November 2010).
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